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Swapping Colored Balls from Bags

Q: Alice has a bag with two red balls and a blue ball. There is another bag with two blue balls and one red ball. A ball is chosen randomly from Alice's bag and placed in the other bag. Next a random ball is chosen from that other bag and put back in Alice's bag. What is the probability that the remainder of the balls in her bag are red if

If the ball Alice draws is blue, there are only two scenarios that could have caused this.

A red ball was drawn at the first draw from the bag and a red ball returned.

A blue ball was drawn at the first draw from the bag and a blue ball returned.

Note that any other scenario would not result in two red balls to remain in the bag while simultaneously being able to draw a blue ball. The probability of scenario 1) is \(\frac{2}{3}\times\frac{1}{2}\times\frac{1}{3}=\frac{1}{9}\) and the probability of scenario 2) is \(\frac{1}{3}\times\frac{3}{4}\times\frac{1}{3}=\frac{1}{12}\). As the scenarios are independent, the sought probability is \(\frac{1}{9} + \frac{1}{12} = \frac{7}{36}\)

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.